Table Of ContentFractals: Theory and Applications in Engineering
Springer
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Michel Dekking, Jacques Levy Vehel,
Evelyne Lutton and Claude Tricot
(Eds.)
Fractals:
Theory and Applications in Engineering
i
Springer
MichelDekking,Professor
FacultyofTechnicalMathematicsausSC,DelftUniversityofTechnology,
Melalweg4,2628CDDelft,TheNetherlands
JacquesLevyWhel,Doctor
INRIA,Rocquencourt,B.P. 105,78153LeChesnayCedex,France
EvelyneLutton,Doctor
INRIA,Rocquencourt,B.P. 105,78153LeChesnayCedex,France
ClaudeTricot,Professor
UniversiteBlaisePascal,DepartementMathematiques,63177AubiereCedex,France
ISBN-13:978-1-4471-1225-9 Springer-VerlagLondonBerlinHeidelberg
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Foreword
The activity around the fractal models develops itselfso rapidly that it is neces
sary, at regulartimes, tosketcha surveyofthe new applicationsand discoveries.
Someoldtopicsseemto cometoa kind ofmaturity, likea localmaximum; whilst
othertopics springon a positiveslope, bringing discussions and dynamics to the
subject. The readers of the former book Fractals in Engineering; From Theory
to Industrial Applications (Springer-Verlag 1997, ed. J. Levy-Vehel, E. Lutton
and C. Tricot) may find after two yearssome major differences with the present
collection ofworks.
At first, a general impression is the following: Mathematics are more and
more involved in the definition and use of fractal models. Let us mention two
particularly active areas which both are strongly based upon theoretical argu
ments. Firstly, stochastic processes defined as extensions offractional Brownian
Motion functions become more and more sophisticated and adaptable to many
experimental situations. Secondly, multifractal analysis has proved itselfa pow
erful and versatile technique in Signal Processing. In both directions, new tools
and new theorems are found regularly and the field ofapplications continues to
grow.
The readers will find consequently several chapters on fractal stochastic pro
cesses in this book. Multifractal spectraare used in texture analysis and several
new models are proposed. Some new achievements on IFS theory and image
compression are given. Wavelets occur in different places (processes, fractal lat
tices,compression)and continuetoplaytheirfundamental roleinsignalanalysis.
Some intriguing vector calculus on fractal curves is sketched: This may well be
a major topic in the near future. The chapters on fractal pores, fractal tunnels,
chaoticflows and monolayerstructures are good representativesofthe wealth of
activity occuring in physics and chemistry in connection with fractals. Note in
particular a paper involving conformal invariance, quantum gravity and multi
fractal analysisthat bringsnew insightsinto percolationphenomenaand, among
other results, proves the famous 4/3 Mandelbrot's conjecture on the boundary
ofa Brownian trajectory.
Finally, it seems that the time when the field was mainly concerned with
a qualitative observation of fractal phenomena has definitely gone. In order to
VI
get operational and closer to the real world, the models are now deep-rooted in
mathematics, with new theory behind.
We wish to thank all the authors who have generously contributed through
recent and original work to the composition ofthis book. We also thank Mitzi
Adams, PierreAdler, AntoineAyache,ChristopheCanus,MarcChassery, Nathan
Cohen, Kenneth Falconer, Bertrand Guiheneuf, Stephane Jaffard, Georges Op
penheim, JacquesPeyriere,PeterPfeifer, MichelRosso,for theirhelp andcareful
reviews, and Nathalie Gaudechoux, for her Latexskill. Oncemore, the efficiency
ofour publisher Springer-Verlag is warmly thanked. We are also deeply grateful
to INRIA and the University ofTechnology ofDelft for their support.
Michel DEKKING,
Jacques LEVY VEHEL,
Evelyne LUTTON,
Claude TRICOT.
Table of Contents
LOCALLY SELF SIMILAR PROCESSES
From Self-Similarity to Local Self-Similarity: the Estimation Problem. .... 3
Serge Cohen
Generalized Multifractional Brownian Motion: Definition and Preliminary
Results.. .. .. .... .. .. .. .. ....... .. .. ..... .... 17
Antoine Ayache, Jacques Levy Vehel
Elliptic SelfSimilar Stochastic Processes. ............................. 33
Albert Benassi, Daniel Roux
Wavelets for Scaling Processes. ...................................... 47
Patrick Flandrin, Patrice Abry
MULTIFRACTAL ANALYSIS
Classification of Natural Texture Images from Shape Analysis ofthe Leg
endre Multifractal Spectrum. ........................................ 67
Piotr Stanczyk, Peter Sharpe
AGeneralizationofMultifractalAnalysisBasedonPolynomialExpansions
ofthe Generating Function. ......................................... 81
Antoine Saucier, Jiri Muller
Local Effective Holder Exponent Estimation on the Wavelet Transform
Maxima Tree ...................................................... 93
Zbigniew R. Struzik
Easy and Natural Generation of Multifractals: Multiplying Harmonics of
Periodic Functions 113
Marc-Olivier Coppens, Benoit B. Mandelbrot
IFS
IFS-Type Operators on Integral Transforms 125
Bruno Forte, Franklin Mendivil, Edward R. Vrscay
Comparison ofDimensions ofa Self-Similar Attractor 139
Serge Dubuc, Jun Li
VIII
FRACTIONAL CALCULUS
Vector Analysis on Fractal Curves 155
Massimiliano Giona
Local Fractional Calculus: a Calculus for Fractal Space-Time 171
Kiran M. Kolwankar, Anil D. Gangal
PHYSICAL SCIENCES
Conformal Multifractality of Random Walks, Polymers, and Percolation
in Two Dimensions 185
Bertrand Duplantier
FractalPoresand FractalTunnels: Trapsfor "Particles" or "SoundParticles"207
Jerome Dorignac, Bernard Sapoval
Fractal Pores and the Degradation ofShales 229
Luis E. Vallejo, Ann Stewart Murphy
Continuous Wavelet Transform Analysis ofFractal Superlattices 245
Herve Aubert, Dwight L. Jaggard
CHEMICAL ENGINEERING
MixinginLaminarChaoticFlows: DifferentiableStructuresand Multifrac-
tal Features 263
Massimiliano Giona
Adhesion AFM Applied to Lipid Monolayers. A Fractal Analysis. . 277
Gianina Dobrescu, Camelia Obreja, Mircea Rusu
IMAGE COMPRESSION
Faster Fractal Image Coding Using Similarity Search in a KL-transformed
Feature Space 293
Jean Cardinal
Can One Break the "Collage Barrier" in Fractal Image Coding? 307
Edward R. Vrscay, Dietmar Saupe
Two Algorithms for Non-Separable Wavelet Transforms and Applications
to Image Compression 325
Franklin Mendivil, Daniel Piche
LOCALLY SELF SIMILAR PROCESSES
From Self-Similarity to Local Self-Similarity:
the Estimation Problem
Serge Cohen
Universite de Versailles-St Quentin en Yvelines
45, avenue des Etats-Unis, 78035 Versailles FRANCE
or
CERMICS-ENPC
cohen~ath.uvsq.fr
Abstract. In this article we review some methods used to identify the
orderH ofafractional Brownianmotion.Thisdiscussion isintroducedto
seehowsuchtechniquescan beextendedtolocallyself-similar processes.
Moreover the model ofthe multifractional Brownian motion which is a
locally self-similar process is further studied. In particular it is shown
that its presentation given by J. Levy Vehel and R. Peltier and the one
given by A. Benassi, S. Jaffard and D. Roux are in some sense Fourier
transformed of each other. Last some results for the estimation of the
multifractionnal Brownian motion are recalled.
Introduction
It is now a classical technique to estimate the scalar index that is relevant to
describe the self-similarity property of a Gaussian process. Let us recall that
in the Gaussian framework there is only one self-similar Gaussian process with
stationary increments (see also [8] for a complete discussion of stationary self
similar Gaussian random fields defined in a generalized sense): the fractional
Brownian motion (fBm) of index H (0 < H < 1) which can be defined by
Xo=0 a.s. and lE(Xt - Xs)2 =It- SI2H. By computing the covariance of the
fBm one can see easily that:
VA> 0 (1)
where (!j means that for every (t1,... ,tn) the distribution of (X-xtll'" ,X-XtJ
is thesameas thoseofAH(Xtl,... ,XtJ. This means that thefBm is self-similar
with index H. Estimating the parameter H by experimental data is a classical
but major issue for applications, and various methods has been proposed.
Actually the parameter H governs at least two other properties of the fBm
beside the self-similarity and I will classify these techniques by stressing which
property is used for the estimation. Moreover the aim ofthis classification is to
discuss whether these techniques can also be applied to locally self-similar pro
cesses. Theneedfor localizingtheself-similarity (1) is motivated by applications.
